I don't think they will let you stay at the Regan bldg. Security is tight. Metal detectors and guards. Depending on the time of day you deliver, L'enfant Promenade off of Independence Ave. SW is a pretty good to park. It is right in front of the "castle" of the Smithsonian.
Turn off of Independence onto L'enfant. Go up about 3 or 4 blocks just past the Postal bldg. (not a post office though) and make a U turn at the end of the median. Don't worry you can do it in a straight truck. Plenty of room and not much traffic on this street. Then park. Straight ahead is the Smithsonian. Turn right to go to Air & Space, Natural and American history museums. Go straight behind the Castle and you are in the "mall". Turn right to go to the Capital and White House. Turn left to go to Washington Monument, Lincoln Memorial, WWII, Vietnam and Korean war memorials.
By the way, you will be parked in front of L'enfant Plaza Metro. So you could take the Metro over to Arlington, if you do the women's armed services museum is right at the entrance to the cemetery where the metro stop is. Or you could take the metro to Georgetown, Chinatown or Penn Station. If you go to Penn Station, the Capital is across the street from the main ent.
My wife and I always park there in our car and walk everywhere. It seems like a lot, well it us if you do it in a day. But from where you park you can also walk to the Jefferson Memorial, Mint, Holocaust Museum and FDR Memorial.
We actually spent a month anchored in our boat in the national harbor which is the opposite of the Smithsonian and down a grassy knoll from L'enfant. We walked everywhere. We walked over to Arlington Cemetery. That was a hike. We took the Metro back.
Note of caution: security has gotten really really tight in DC. I have never parked my truck there but have seen trucks parked there. No parking meters, free parking. Once there was guards at the entrance to L'enfant, but I think they were for the two gov. Bldgs on the corner of L'enfant and Independence.
Like I said, we always park there when we go to DC and play tourist, never parked there over night, tho. I don't think it would be a problem cause that part of town is a ghost town after about 10 PM and safe.
Hope it works out. Don't get mad at me if it don't!